Finding Y When X=5 In The Equation Y=(x-3)^2+1

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Finding y When x=5 in the Equation y=(x-3)^2+1

Let's dive into solving this math problem, guys! We're given the equation y=(x-3)^2+1 and we need to figure out what y is when x is equal to 5. It might seem a bit daunting at first, but trust me, it's super straightforward once we break it down step by step. So, grab your pencils, your thinking caps, and let's get started!

Understanding the Equation

First, let's take a good look at the equation we're working with: y=(x-3)^2+1. This is a quadratic equation, and it represents a parabola. Don't let those terms scare you! A parabola is just a U-shaped curve, and quadratic equations are used to describe them. The key here is understanding what each part of the equation means. We have y, which is our dependent variable – its value depends on what x is. Then we have x, which is our independent variable – we get to choose its value. The equation tells us exactly how to calculate y once we know x. We subtract 3 from x, square the result, and then add 1. This might seem like a bunch of abstract stuff, but it's actually a recipe for finding y! So, when dealing with equations like these, the first step is always to make sure you understand what each variable represents and how they relate to each other. This foundational understanding will make the rest of the problem much easier to tackle. Remember, math is like building with LEGOs – you need to understand the basic blocks before you can build something complex.

Substituting x = 5 into the Equation

The heart of solving this problem lies in substitution. We're told that x=5, and our mission is to find the corresponding value of y. Substitution simply means replacing the variable x in the equation with the number 5. It’s like swapping out a piece in a puzzle. So, let's do it! Our equation, y=(x-3)^2+1, becomes y=(5-3)^2+1. See what we did? We just replaced the x with a 5. Now, the equation looks much simpler, doesn't it? We've transformed it from an equation with two variables into an equation with just one variable, y. This is a crucial step because now we can directly solve for y. Remember, the goal in most math problems is to isolate the variable you're trying to find. Substitution is a powerful tool that helps us do just that. It's like having a secret code that allows us to unlock the value of y. So, never underestimate the power of substitution! It's a fundamental technique in algebra and a skill you'll use again and again.

Order of Operations: A Quick Refresher

Before we jump into calculating the value of y, let's quickly review the order of operations. This is a super important concept in math that tells us the correct sequence to perform mathematical operations. Remember the acronym PEMDAS or BODMAS? It stands for:

  • Parentheses (or Brackets)
  • Exponents (or Orders)
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

This order is crucial because doing operations in the wrong order will lead to the wrong answer. Think of it like following a recipe – if you add the ingredients in the wrong order, you might end up with a cake that doesn't taste so great! In our equation, y=(5-3)^2+1, we have parentheses, an exponent, and addition. So, according to PEMDAS/BODMAS, we need to deal with the parentheses first, then the exponent, and finally the addition. Keeping the order of operations in mind will ensure that we arrive at the correct solution. It's like having a roadmap that guides us through the calculation process.

Step-by-Step Calculation

Alright, let's put our knowledge of the order of operations to work and calculate the value of y. We have the equation y=(5-3)^2+1. Following PEMDAS/BODMAS, we start with the parentheses:

  1. (5-3) = 2

So, our equation now becomes y=(2)^2+1. Next up, we deal with the exponent:

  1. (2)^2 = 2 * 2 = 4

Now our equation is even simpler: y=4+1. Finally, we perform the addition:

  1. 4 + 1 = 5

And there you have it! We've successfully calculated the value of y. By following the order of operations step-by-step, we've transformed a seemingly complex equation into a simple arithmetic problem. Each step built upon the previous one, leading us to the final answer. This methodical approach is key to solving mathematical problems. It's like climbing a ladder – you take it one step at a time until you reach the top.

The Final Answer

After carefully substituting x=5 into the equation y=(x-3)^2+1 and following the order of operations, we've arrived at our final answer: y=5. This means that when x is 5, the value of y is also 5. It's like finding a matching pair – when x is 5, its partner y is also 5. This solution represents a specific point on the parabola defined by the equation. If we were to graph the equation, the point (5, 5) would lie on the curve. So, we've not only found the value of y, but we've also gained a little insight into the behavior of the equation. This is the beauty of mathematics – it's not just about finding numbers, it's about understanding the relationships between them. And in this case, we've successfully uncovered the relationship between x and y when x is 5.

Conclusion

So, guys, we've successfully solved for y when x=5 in the equation y=(x-3)^2+1. We walked through the process step-by-step, emphasizing the importance of understanding the equation, using substitution, remembering the order of operations, and performing the calculations carefully. We found that when x is 5, y is also 5. This problem might seem simple, but it illustrates some fundamental concepts in algebra that are crucial for tackling more complex problems. Remember, math is like building a house – you need a strong foundation before you can build the walls and the roof. By mastering these basic skills, you'll be well-equipped to conquer any mathematical challenge that comes your way. Keep practicing, keep exploring, and most importantly, keep having fun with math!